# Properties

 Label 144.2.x.c Level $144$ Weight $2$ Character orbit 144.x Analytic conductor $1.150$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,2,Mod(13,144)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 9, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("144.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 144.x (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.14984578911$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} - 2 \zeta_{12} q^{4} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2}) q^{5} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} + 1) q^{6} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 2) q^{7} + (2 \zeta_{12}^{3} - 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} +O(q^{10})$$ q + (-z^3 - z^2 + z) * q^2 + (z^3 + z) * q^3 - 2*z * q^4 + (-2*z^3 + 2*z^2) * q^5 + (-2*z^3 + z^2 + z + 1) * q^6 + (-z^3 - z^2 + z + 2) * q^7 + (2*z^3 - 2) * q^8 + (3*z^2 - 3) * q^9 $$q + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} - 2 \zeta_{12} q^{4} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2}) q^{5} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} + 1) q^{6} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 2) q^{7} + (2 \zeta_{12}^{3} - 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} + (2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{10} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - \zeta_{12} + 3) q^{11} + ( - 4 \zeta_{12}^{2} + 2) q^{12} + (4 \zeta_{12}^{3} + \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{13} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2}) q^{14} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{15} + 4 \zeta_{12}^{2} q^{16} + (\zeta_{12}^{3} - 2 \zeta_{12} - 4) q^{17} + (3 \zeta_{12}^{3} + 3) q^{18} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} - 2) q^{19} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4) q^{20} + (\zeta_{12}^{2} + 3 \zeta_{12} + 1) q^{21} + ( - 3 \zeta_{12}^{2} - \zeta_{12} - 3) q^{22} + ( - \zeta_{12}^{2} - 3 \zeta_{12} - 1) q^{23} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 4) q^{24} + ( - 3 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 3 \zeta_{12} - 8) q^{25} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} - 2) q^{26} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + (2 \zeta_{12}^{3} - 4 \zeta_{12} - 2) q^{28} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 2) q^{29} + ( - 6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 6 \zeta_{12} - 4) q^{30} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 2 \zeta_{12} - 4) q^{31} + ( - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{32} + ( - \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 5 \zeta_{12} + 5) q^{33} + (5 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 3 \zeta_{12} - 2) q^{34} + ( - 2 \zeta_{12}^{3} + 2) q^{35} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{36} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{37} + (\zeta_{12}^{3} + 3 \zeta_{12}^{2} + \zeta_{12}) q^{38} + (5 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} - 5) q^{39} + (8 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{40} - 3 \zeta_{12} q^{41} + ( - 4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 4) q^{42} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 2 \zeta_{12} + 3) q^{43} + (4 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12} - 4) q^{44} + (6 \zeta_{12} - 6) q^{45} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 4) q^{46} + (5 \zeta_{12}^{3} - \zeta_{12}^{2} + 5 \zeta_{12}) q^{47} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{48} + ( - 4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 2 \zeta_{12} - 3) q^{49} + (8 \zeta_{12}^{3} + \zeta_{12}^{2} - 7 \zeta_{12} + 7) q^{50} + ( - 4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 4 \zeta_{12}) q^{51} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 6 \zeta_{12} + 8) q^{52} + ( - 2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12} - 2) q^{53} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12} - 6) q^{54} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{55} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4) q^{56} + ( - 3 \zeta_{12} - 3) q^{57} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{58} + ( - \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 5 \zeta_{12} + 5) q^{59} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 8 \zeta_{12} + 8) q^{60} + (6 \zeta_{12} + 6) q^{61} + (6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 6) q^{62} + (3 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{63} - 8 \zeta_{12}^{3} q^{64} + ( - 6 \zeta_{12}^{3} + 14 \zeta_{12}^{2} - 6 \zeta_{12}) q^{65} + ( - 9 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{66} + ( - 5 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{67} + (2 \zeta_{12}^{2} + 8 \zeta_{12} + 2) q^{68} + ( - 3 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{69} - 4 \zeta_{12}^{2} q^{70} + (4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{71} + ( - 6 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{72} + (8 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{73} + ( - 6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 6 \zeta_{12} + 4) q^{74} + (3 \zeta_{12}^{2} - 12 \zeta_{12} + 3) q^{75} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{76} + ( - 4 \zeta_{12}^{3} - 7 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{77} + ( - 2 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 2 \zeta_{12}) q^{78} - 12 \zeta_{12}^{2} q^{79} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 8 \zeta_{12} - 8) q^{80} - 9 \zeta_{12}^{2} q^{81} + (3 \zeta_{12}^{3} - 3) q^{82} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{83} + ( - 2 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 2 \zeta_{12}) q^{84} + (6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{85} + ( - 3 \zeta_{12}^{2} - 7 \zeta_{12} - 3) q^{86} + (3 \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{87} + (6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 6 \zeta_{12}) q^{88} + 2 \zeta_{12}^{3} q^{89} + (6 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{90} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} + 4) q^{91} + (2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 2 \zeta_{12}) q^{92} + (4 \zeta_{12}^{3} - 8 \zeta_{12} + 6) q^{93} + ( - 10 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{94} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{95} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 8 \zeta_{12} - 4) q^{96} + ( - \zeta_{12}^{3} - 10 \zeta_{12}^{2} - \zeta_{12}) q^{97} + (5 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 5) q^{98} + ( - 3 \zeta_{12}^{3} + 9 \zeta_{12}^{2} + 9 \zeta_{12} - 3) q^{99} +O(q^{100})$$ q + (-z^3 - z^2 + z) * q^2 + (z^3 + z) * q^3 - 2*z * q^4 + (-2*z^3 + 2*z^2) * q^5 + (-2*z^3 + z^2 + z + 1) * q^6 + (-z^3 - z^2 + z + 2) * q^7 + (2*z^3 - 2) * q^8 + (3*z^2 - 3) * q^9 + (2*z^3 - 4*z^2 + 2) * q^10 + (-2*z^3 - 2*z^2 - z + 3) * q^11 + (-4*z^2 + 2) * q^12 + (4*z^3 + z^2 - 3*z + 3) * q^13 + (-2*z^3 - 2*z^2) * q^14 + (4*z^3 - 2*z^2 - 2*z + 4) * q^15 + 4*z^2 * q^16 + (z^3 - 2*z - 4) * q^17 + (3*z^3 + 3) * q^18 + (2*z^3 + z^2 - z - 2) * q^19 + (-4*z^3 + 4*z^2 - 4) * q^20 + (z^2 + 3*z + 1) * q^21 + (-3*z^2 - z - 3) * q^22 + (-z^2 - 3*z - 1) * q^23 + (-2*z^3 + 2*z^2 - 2*z - 4) * q^24 + (-3*z^3 + 4*z^2 + 3*z - 8) * q^25 + (-4*z^3 + 8*z - 2) * q^26 + (3*z^3 - 6*z) * q^27 + (2*z^3 - 4*z - 2) * q^28 + (z^3 - z^2 + z + 2) * q^29 + (-6*z^3 + 2*z^2 + 6*z - 4) * q^30 + (-4*z^3 + 4*z^2 + 2*z - 4) * q^31 + (-4*z^2 + 4*z + 4) * q^32 + (-z^3 - 4*z^2 + 5*z + 5) * q^33 + (5*z^3 + 5*z^2 - 3*z - 2) * q^34 + (-2*z^3 + 2) * q^35 + (-6*z^3 + 6*z) * q^36 + (2*z^3 + 2*z^2 + 2*z + 2) * q^37 + (z^3 + 3*z^2 + z) * q^38 + (5*z^3 - 2*z^2 + 2*z - 5) * q^39 + (8*z^3 - 4*z^2 - 4*z + 4) * q^40 - 3*z * q^41 + (-4*z^3 - 2*z^2 + 2*z + 4) * q^42 + (-5*z^3 - 5*z^2 + 2*z + 3) * q^43 + (4*z^3 + 6*z^2 - 6*z - 4) * q^44 + (6*z - 6) * q^45 + (4*z^3 + 2*z^2 - 2*z - 4) * q^46 + (5*z^3 - z^2 + 5*z) * q^47 + (8*z^3 - 4*z) * q^48 + (-4*z^3 + 3*z^2 + 2*z - 3) * q^49 + (8*z^3 + z^2 - 7*z + 7) * q^50 + (-4*z^3 - 3*z^2 - 4*z) * q^51 + (-2*z^3 - 2*z^2 - 6*z + 8) * q^52 + (-2*z^3 - 4*z^2 - 4*z - 2) * q^53 + (3*z^3 + 3*z^2 + 3*z - 6) * q^54 + (-8*z^3 + 4*z^2 - 2) * q^55 + (4*z^3 + 4*z^2 - 4) * q^56 + (-3*z - 3) * q^57 + (-4*z^3 + 2*z) * q^58 + (-z^3 - 4*z^2 + 5*z + 5) * q^59 + (4*z^3 - 4*z^2 - 8*z + 8) * q^60 + (6*z + 6) * q^61 + (6*z^3 - 4*z^2 - 4*z + 6) * q^62 + (3*z^3 + 6*z^2 - 3) * q^63 - 8*z^3 * q^64 + (-6*z^3 + 14*z^2 - 6*z) * q^65 + (-9*z^3 - 2*z^2 + 1) * q^66 + (-5*z^3 - 3*z^2 + 2*z - 2) * q^67 + (2*z^2 + 8*z + 2) * q^68 + (-3*z^3 - 6*z^2 + 3) * q^69 - 4*z^2 * q^70 + (4*z^3 - 8*z^2 + 4) * q^71 + (-6*z^2 - 6*z + 6) * q^72 + (8*z^3 - 2*z^2 + 1) * q^73 + (-6*z^3 - 2*z^2 + 6*z + 4) * q^74 + (3*z^2 - 12*z + 3) * q^75 + (-2*z^3 - 2*z^2 + 4*z + 4) * q^76 + (-4*z^3 - 7*z^2 - 3*z + 3) * q^77 + (-2*z^3 + 12*z^2 - 2*z) * q^78 - 12*z^2 * q^79 + (-8*z^3 + 8*z^2 + 8*z - 8) * q^80 - 9*z^2 * q^81 + (3*z^3 - 3) * q^82 + (z^3 - z^2 - z) * q^83 + (-2*z^3 - 6*z^2 - 2*z) * q^84 + (6*z^3 - 4*z^2 - 2*z - 2) * q^85 + (-3*z^2 - 7*z - 3) * q^86 + (3*z^2 + 3*z - 3) * q^87 + (6*z^3 + 2*z^2 + 6*z) * q^88 + 2*z^3 * q^89 + (6*z^2 - 6*z + 6) * q^90 + (4*z^3 + 2*z^2 + 2*z + 4) * q^91 + (2*z^3 + 6*z^2 + 2*z) * q^92 + (4*z^3 - 8*z + 6) * q^93 + (-10*z^3 + 6*z^2 + 4*z + 4) * q^94 + (4*z^3 - 2*z) * q^95 + (-4*z^3 + 8*z^2 + 8*z - 4) * q^96 + (-z^3 - 10*z^2 - z) * q^97 + (5*z^3 - 4*z^2 - 4*z + 5) * q^98 + (-3*z^3 + 9*z^2 + 9*z - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 4 q^{5} + 6 q^{6} + 6 q^{7} - 8 q^{8} - 6 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 + 4 * q^5 + 6 * q^6 + 6 * q^7 - 8 * q^8 - 6 * q^9 $$4 q - 2 q^{2} + 4 q^{5} + 6 q^{6} + 6 q^{7} - 8 q^{8} - 6 q^{9} + 8 q^{11} + 14 q^{13} - 4 q^{14} + 12 q^{15} + 8 q^{16} - 16 q^{17} + 12 q^{18} - 6 q^{19} - 8 q^{20} + 6 q^{21} - 18 q^{22} - 6 q^{23} - 12 q^{24} - 24 q^{25} - 8 q^{26} - 8 q^{28} + 6 q^{29} - 12 q^{30} - 8 q^{31} + 8 q^{32} + 12 q^{33} + 2 q^{34} + 8 q^{35} + 12 q^{37} + 6 q^{38} - 24 q^{39} + 8 q^{40} + 12 q^{42} + 2 q^{43} - 4 q^{44} - 24 q^{45} - 12 q^{46} - 2 q^{47} - 6 q^{49} + 30 q^{50} - 6 q^{51} + 28 q^{52} - 16 q^{53} - 18 q^{54} - 8 q^{56} - 12 q^{57} + 12 q^{59} + 24 q^{60} + 24 q^{61} + 16 q^{62} + 28 q^{65} - 14 q^{67} + 12 q^{68} - 8 q^{70} + 12 q^{72} + 12 q^{74} + 18 q^{75} + 12 q^{76} - 2 q^{77} + 24 q^{78} - 24 q^{79} - 16 q^{80} - 18 q^{81} - 12 q^{82} - 2 q^{83} - 12 q^{84} - 16 q^{85} - 18 q^{86} - 6 q^{87} + 4 q^{88} + 36 q^{90} + 20 q^{91} + 12 q^{92} + 24 q^{93} + 28 q^{94} - 20 q^{97} + 12 q^{98} + 6 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 + 4 * q^5 + 6 * q^6 + 6 * q^7 - 8 * q^8 - 6 * q^9 + 8 * q^11 + 14 * q^13 - 4 * q^14 + 12 * q^15 + 8 * q^16 - 16 * q^17 + 12 * q^18 - 6 * q^19 - 8 * q^20 + 6 * q^21 - 18 * q^22 - 6 * q^23 - 12 * q^24 - 24 * q^25 - 8 * q^26 - 8 * q^28 + 6 * q^29 - 12 * q^30 - 8 * q^31 + 8 * q^32 + 12 * q^33 + 2 * q^34 + 8 * q^35 + 12 * q^37 + 6 * q^38 - 24 * q^39 + 8 * q^40 + 12 * q^42 + 2 * q^43 - 4 * q^44 - 24 * q^45 - 12 * q^46 - 2 * q^47 - 6 * q^49 + 30 * q^50 - 6 * q^51 + 28 * q^52 - 16 * q^53 - 18 * q^54 - 8 * q^56 - 12 * q^57 + 12 * q^59 + 24 * q^60 + 24 * q^61 + 16 * q^62 + 28 * q^65 - 14 * q^67 + 12 * q^68 - 8 * q^70 + 12 * q^72 + 12 * q^74 + 18 * q^75 + 12 * q^76 - 2 * q^77 + 24 * q^78 - 24 * q^79 - 16 * q^80 - 18 * q^81 - 12 * q^82 - 2 * q^83 - 12 * q^84 - 16 * q^85 - 18 * q^86 - 6 * q^87 + 4 * q^88 + 36 * q^90 + 20 * q^91 + 12 * q^92 + 24 * q^93 + 28 * q^94 - 20 * q^97 + 12 * q^98 + 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$\zeta_{12}^{3}$$ $$-1 + \zeta_{12}^{2}$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
−1.36603 0.366025i −0.866025 1.50000i 1.73205 + 1.00000i 1.00000 + 3.73205i 0.633975 + 2.36603i 0.633975 0.366025i −2.00000 2.00000i −1.50000 + 2.59808i 5.46410i
61.1 0.366025 + 1.36603i 0.866025 1.50000i −1.73205 + 1.00000i 1.00000 + 0.267949i 2.36603 + 0.633975i 2.36603 + 1.36603i −2.00000 2.00000i −1.50000 2.59808i 1.46410i
85.1 0.366025 1.36603i 0.866025 + 1.50000i −1.73205 1.00000i 1.00000 0.267949i 2.36603 0.633975i 2.36603 1.36603i −2.00000 + 2.00000i −1.50000 + 2.59808i 1.46410i
133.1 −1.36603 + 0.366025i −0.866025 + 1.50000i 1.73205 1.00000i 1.00000 3.73205i 0.633975 2.36603i 0.633975 + 0.366025i −2.00000 + 2.00000i −1.50000 2.59808i 5.46410i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
144.x even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.2.x.c yes 4
3.b odd 2 1 432.2.y.b 4
4.b odd 2 1 576.2.bb.c 4
9.c even 3 1 144.2.x.b 4
9.d odd 6 1 432.2.y.c 4
12.b even 2 1 1728.2.bc.a 4
16.e even 4 1 144.2.x.b 4
16.f odd 4 1 576.2.bb.d 4
36.f odd 6 1 576.2.bb.d 4
36.h even 6 1 1728.2.bc.d 4
48.i odd 4 1 432.2.y.c 4
48.k even 4 1 1728.2.bc.d 4
144.u even 12 1 1728.2.bc.a 4
144.v odd 12 1 576.2.bb.c 4
144.w odd 12 1 432.2.y.b 4
144.x even 12 1 inner 144.2.x.c yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.x.b 4 9.c even 3 1
144.2.x.b 4 16.e even 4 1
144.2.x.c yes 4 1.a even 1 1 trivial
144.2.x.c yes 4 144.x even 12 1 inner
432.2.y.b 4 3.b odd 2 1
432.2.y.b 4 144.w odd 12 1
432.2.y.c 4 9.d odd 6 1
432.2.y.c 4 48.i odd 4 1
576.2.bb.c 4 4.b odd 2 1
576.2.bb.c 4 144.v odd 12 1
576.2.bb.d 4 16.f odd 4 1
576.2.bb.d 4 36.f odd 6 1
1728.2.bc.a 4 12.b even 2 1
1728.2.bc.a 4 144.u even 12 1
1728.2.bc.d 4 36.h even 6 1
1728.2.bc.d 4 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 4T_{5}^{3} + 20T_{5}^{2} - 32T_{5} + 16$$ acting on $$S_{2}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$T^{4} - 4 T^{3} + 20 T^{2} - 32 T + 16$$
$7$ $$T^{4} - 6 T^{3} + 14 T^{2} - 12 T + 4$$
$11$ $$T^{4} - 8 T^{3} + 41 T^{2} - 130 T + 169$$
$13$ $$T^{4} - 14 T^{3} + 74 T^{2} + \cdots + 484$$
$17$ $$(T^{2} + 8 T + 13)^{2}$$
$19$ $$T^{4} + 6 T^{3} + 18 T^{2} + 18 T + 9$$
$23$ $$T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36$$
$29$ $$T^{4} - 6 T^{3} + 18 T^{2} - 36 T + 36$$
$31$ $$T^{4} + 8 T^{3} + 60 T^{2} + 32 T + 16$$
$37$ $$T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 144$$
$41$ $$T^{4} - 9T^{2} + 81$$
$43$ $$T^{4} - 2 T^{3} + 65 T^{2} + 176 T + 121$$
$47$ $$T^{4} + 2 T^{3} + 78 T^{2} + \cdots + 5476$$
$53$ $$T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 64$$
$59$ $$T^{4} - 12 T^{3} + 45 T^{2} + \cdots + 1521$$
$61$ $$T^{4} - 24 T^{3} + 180 T^{2} + \cdots + 1296$$
$67$ $$T^{4} + 14 T^{3} + 113 T^{2} + \cdots + 1369$$
$71$ $$T^{4} + 128T^{2} + 1024$$
$73$ $$T^{4} + 134T^{2} + 3721$$
$79$ $$(T^{2} + 12 T + 144)^{2}$$
$83$ $$T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4$$
$89$ $$(T^{2} + 4)^{2}$$
$97$ $$T^{4} + 20 T^{3} + 303 T^{2} + \cdots + 9409$$